Many operating processes have varying dynamics characteristics which are notoriously difficult to model and control. These processes are extremely diverse and can be found in virtually any field of endeavor. One example of such operating processes is particle accelerators used to study fundamental particles. The study of fundamental particles and their interactions seeks to answer two questions: (1) what are the fundamental building blocks (smallest) from which all matter is made; and (2) what are the interactions between these particles that govern how the particles combine and decay? To answer these questions, physicists use accelerators to provide high energy to subatomic particles, which then collide with targets. Out of these interactions come many other subatomic particles that pass into detectors. FIGS. 1A and 1B illustrate typical collisions or interactions used in this study. From the information gathered in the detectors, physicists can determine properties of the particles and their interactions.
In these experiments, subatomic particles collide. However, to achieve the desired experiments requires a large degree of control over the particles trajectory and the environment in which the collisions actually take place. Process and control models are typically used to aid the physicist in the setup and execution of these experiments.
Process Models used for prediction, control, and optimization can be divided into two general categories, steady state models and dynamic models. These models are mathematical constructs that characterize the process, and process measurements are often utilized to build these mathematical constructs in a way that the model replicates the behavior of the process. These models can then be used for prediction, optimization, and control of the process.
Many modern process control systems use steady-state or static models. These models often capture the information contained in large amounts of data, wherein this data typically contains steady-state information at many different operating conditions. In general, the steady-state model is a non-linear model wherein the process input variables are represented by the vector U that is processed through the model to output the dependent variable Y. The non-linear steady-state model is a phenomenological or empirical model that is developed utilizing several ordered pairs (Ui, Yi) of data from different measured steady states. If a model is represented as:Y=P(U,Y)   (1)                where P is an appropriate static mapping, then the steady-state modeling procedure can be presented as:M(Ū,{overscore (Y)})→P   (2)        where U and Y are vectors containing the Ui, Yi ordered pair elements. Given the model P, then the steady-state process gain can be calculated as:        
                    K        =                              Δ            ⁢                                                  ⁢                          P              ⁡                              (                                  u                  ,                  y                                )                                                          Δ            ⁢                                                  ⁢            u                                              (        3        )            
The steady-state model, therefore, represents the process measurements taken when the process is in a “static” mode. These measurements do not account for process behavior under non-steady-state condition (e.g. when the process is perturbed, or when process transitions from one steady-state condition to another steady-state condition). It should be noted that real world processes (e.g. particle accelerators, chemical plants) operate within an inherently dynamic environment. Hence steady-state models alone are, in general, not sufficient for prediction, optimization, and control of an inherently dynamic process.
A dynamic model is typically a model obtained from non-steady-state process measurements. These non-steady-state process measurements are often obtained as the process transitions from one steady-state condition to another. In this procedure, process inputs (manipulated and/or disturbance variables denoted by vector u(t)), applied to a process affect process outputs (controlled variables denoted by vector y(t)), that are being output and measured. Again, ordered pairs of measured data (u(ti), y(ti)) represent a phenomenological or empirical model, wherein in this instance data comes from non-steady-state operation. The dynamic model is represented as:y(t)=p(u(t), u(t−1), . . . , u(t−M), y(t), y(t−1), . . . , y(t−N))   (4)                where p is an appropriate mapping. M and N specify the input and output history that is required to build the dynamic model.The state-space description of a dynamic system is equivalent to input/output description in Equation (4) for appropriately chosen M and N values, and hence the description in Equation (4) encompasses state-space description of the dynamic systems/processes as well.        
Nonlinear dynamic systems are in general difficult to build. Prior art includes a variety of model structures in which a nonlinear static model and a linear dynamic model are combined in order to represent a nonlinear dynamic system. Examples include Hammerstein models (where a static nonlinear model precedes a linear dynamic model in a series connection), and Wiener models (where a linear dynamic model precedes a static nonlinear model in a series connection). U.S. Pat. No. 5,933,345 constructs a nonlinear dynamic model in which the nonlinear model respects the nonlinear static mapping captured by a neural network.